Measurements of differential cross sections for inclusive very forward jet production in proton-lead collisions as a function of jet energy are presented. The data were collected with the CMS experiment at the LHC in the laboratory pseudorapidity range $-$6.6$<\eta<-$5.2. Asymmetric beam energies of 4 TeV for protons and 1.58 TeV per nucleon for Pb nuclei were used, corresponding to a center-of-mass energy per nucleon pair of $\sqrt{s_\mathrm{NN}}=$ 5.02 TeV. Collisions with either the proton (p+Pb) or the ion (Pb+p) traveling towards the negative $\eta$ hemisphere are studied. The jet cross sections are unfolded to stable-particle level cross sections with $p_\mathrm{T}\gtrsim$ 3 GeV, and compared to predictions from various Monte Carlo ev...